Question: Solve for $x$ : $ 6|x + 9| - 1 = 5|x + 9| + 4 $
Answer: Subtract $ {5|x + 9|} $ from both sides: $ \begin{eqnarray} 6|x + 9| - 1 &=& 5|x + 9| + 4 \\ \\ { - 5|x + 9|} && { - 5|x + 9|} \\ \\ 1|x + 9| - 1 &=& 4 \end{eqnarray} $ Add ${1}$ to both sides: $ \begin{eqnarray} 1|x + 9| - 1 &=& 4 \\ \\ { + 1} &=& { + 1} \\ \\ 1|x + 9| &=& 5 \end{eqnarray} $ Simplify: $ |x + 9| = 5$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x + 9 = -5 $ or $ x + 9 = 5 $ Solve for the solution where $x + 9$ is negative: $ x + 9 = -5 $ Subtract ${9}$ from both sides: $ \begin{eqnarray} x + 9 &=& -5 \\ \\ {- 9} && {- 9} \\ \\ x &=& -5 - 9 \end{eqnarray} $ $ x = -14 $ Then calculate the solution where $x + 9$ is positive: $ x + 9 = 5 $ Subtract ${9}$ from both sides: $ \begin{eqnarray} x + 9 &=& 5 \\ \\ {- 9} && {- 9} \\ \\ x &=& 5 - 9 \end{eqnarray} $ $ x = -4 $ Thus, the correct answer is $x = -14 $ or $x = -4 $.